On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in k-uniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lov´asz [16] gave a k 2 factor LP rounding based approximation, and a matching k 2 − o(1) integrality gap instance was constructed by Aharoni et al. [1]. We prove the following results, which are the first strong hardness results for this problem (here ε > 0 is an arbitrary constant): • NP-hardness of approximating within a factor of k 16 − ε , and • Unique Games-hardness of approximating within a factor of k 2 − ε , showing optimality of Lov´asz’s algorithm under the Unique Games conjecture. The NP-hardness result is based on a reduction fro...

• Type: Conference
• Year: 2010
• Where: ICALP
• Authors: Venkatesan Guruswami, Rishi Saket
• Keywords: Bipartite Vertex Cover, ICALP 2010, Minimum Vertex Cover, Programming Languages, Vertex Cover